if the angles of one triangle are respectively equal to the angles of another triangle prove that the ratio of their corresponding sides is the same ratio is their corresponding angle bisectors
Answers
Step-by-step explanation:
Proved that the ratio of their corresponding sides is the same as the ratio of their corresponding (i) medians (ii)angle bisectors (iii) altitudes.
Step-by-step explanation:
let say ΔABC & ΔPQR
∠A = ∠P
∠B = ∠Q
∠C = R
=> ΔABC ≈ ΔPQR
=> AB/PQ = BC/QR = AC/PR
(i) medians
Lets draw AD & PS median
=> BD = CD = BC/2 & QS = RS = QR/2
BD/QS = (BC/2)/(QR/2) = BC/QR
=> AB/PQ = BD/QS & ∠B = ∠Q
=> ΔABD ≈ ΔPQS
=> AB/PQ = BD/QS
similarly we can show for other medians
(ii)angle bisectors
AD & PS are angle bisectors
=> ∠BAD = ∠A/2 ∠QPS = ∠P/2
=> ∠BAD = ∠QPS
∠B = ∠Q
=> ΔABD ≈ ΔPQS
=> AB/PQ = BD/QS
similarly we can show for other angle bisectors
(ii) altitudes.
AD & PS are altitudes.
=> ∠BDA= 90° ∠QSP = 90°
=> ∠BDA= ∠QSP
∠B = ∠Q
=> ΔABD ≈ ΔPQS
=> AB/PQ = BD/QS
similarly we can show for other altitudes.
Hope it helps!!!